**3. Source and sinks**

The expansion of CFD applications to the fields such as chemical engineering, environmental engineering, thermal engineering and pharmaceutical engineering was based on the development of multiscale and multiphase models which benefited from the coupling of source or sink terms to the conservation equations. A sink term is actually a negative source term, so both source terms and sink terms could be called by a joint name—source terms.

*The Basic Theory of CFD Governing Equations and the Numerical Solution Methods… DOI: http://dx.doi.org/10.5772/intechopen.113253*

#### **3.1 Source term for the continuity equation**

Processes such as solid gasification or combustion, and gas adsorption by solid sorbents would break the mass conservation of the fluid phase. If there are reactions are generating additional fluid or remove the existing fluid in a control volume, the mass balance in this control volume must be disrupted by these reactions. The rate of fluid entering a control volume minus the rate of fluid flowing out equals the rate of fluid accumulating in this control volume and the net gain of fluid generated inside the control volume. If in the control volume there is a reaction generating fluid at a rate of *r*, Eq. (1) for mass balance will be altered to

$$\frac{\text{d}m}{\text{d}t} = r \text{d}x \text{d}y \text{d}z + \sum\_{\text{in}} \dot{m} - \sum\_{\text{out}} \dot{m} \tag{88}$$

Eq. (45) could also include the case where there is a depletion of fluid by using a negative value for *r*. Therefore, the continuity equation is adjusted by a source term as follows.

$$\frac{\partial \rho}{\partial t} + \frac{\partial(\rho u)}{\partial \mathbf{x}} + \frac{\partial(\rho v)}{\partial \mathbf{y}} + \frac{\partial(\rho w)}{\partial \mathbf{z}} = \mathbf{S}\_{\text{continuity}}\tag{89}$$

where the source term is the reaction rate *S*continuity = *r* (kg m�<sup>3</sup> s �1 ) if the reaction is a body reaction. For a surface reaction, the source term is the multiplication of the surface reaction rate (kg m�<sup>2</sup> s �1 ) with the reaction area in unit volume (*A*react/*V*body, m<sup>2</sup> m�<sup>3</sup> ) [10].

#### **3.2 Source term for component equation**

Similar to the imbalance for total mass conservation, the conservation of each component in a multi-component flow with reactions could also be broken. The components participating as reactants would be deemed as sinks and the components generated as products would be regarded as sources. If a component is generated at a rate of *ri*, then Eq. (7) is adjusted accordingly as follows

$$\frac{d\mathbf{m}\_i}{dt} = r\_i \mathbf{dx} \mathbf{d} \mathbf{y} \mathbf{d}z + \sum\_{\text{in}} \dot{m}\_i - \sum\_{\text{out}} \dot{m}\_i \tag{90}$$

After a similar derivation from Eq. (7) to Eq. (9), the component equation should include a source term as expressed by Eq. (91)

$$\begin{split} \frac{\partial(\rho X\_i)}{\partial t} + \frac{\partial(\rho u X\_i)}{\partial \mathbf{x}} + \frac{\partial(\rho v X\_i)}{\partial \mathbf{y}} + \frac{\partial(\rho w X\_i)}{\partial \mathbf{z}} \\ = D \left[ \frac{\partial^2(\rho X\_i)}{\partial \mathbf{x}^2} + \frac{\partial^2(\rho X\_i)}{\partial \mathbf{y}^2} + \frac{\partial^2(\rho X\_i)}{\partial \mathbf{z}^2} \right] + \mathbf{S}\_i \end{split} \tag{91}$$

where *Si* is the source term of the component equation, and it equals reaction rate *ri* if it is a body reaction. If it is a surface reaction, the source term of Eq. (91) equals the multiplication of the surface reaction rate (kg m�<sup>2</sup> s �1 ) with the reaction area in unit volume (*A*react/*V*body, m<sup>2</sup> m�<sup>3</sup> ).

#### **3.3 Source term for the momentum equation**

When it comes to cases with a multiphase flow such as gas-solid flow in fluidized bed, momentum conservation still holds, but interphase interactions need to be taken into account [15]. There might be extra forces between different phases, which can affect the momentum transfer between different phases. Therefore, in multiphase flow, appropriate interphase force terms need to be introduced to correct the momentum equation.

When the fluid phase and solid phase interact with each other, a reactive force that causes flow resistance is generated, known as drag force. If the drag force is expressed as *F*D,i, Eq. (13) can be expanded as follows

$$\frac{\text{Du}}{\text{Dt}} = \frac{\sum F\_{\text{x}}}{m} = \frac{\sum F\_{\text{surf}} + \sum F\_{\text{body}} + \sum F\_{\text{D},i}}{m} \tag{92}$$

The drag force *F*D,I is calculated by:

$$F\_{\rm D,i} = \frac{\beta\_{\rm fs}}{\theta\_{\rm f}} (\mathfrak{u}\_{\rm s} - \mathfrak{u}\_{\rm f}) \tag{93}$$

where *u*<sup>s</sup> is the solid phase velocity, *u*<sup>f</sup> is the fluid phase velocity, *θ*<sup>f</sup> is the volume fraction of the fluid phase and *β*fs is the drag coefficient. Many researchers have proposed the models for drag coefficients. The most widely used is the Gidaspow model which combines Ergun and Wen-Yu equations to accurately simulate the gas-solid multiphase flow:

$$\beta\_{\rm p,f} = \begin{cases} 150 \frac{\theta\_{\rm s}^2 \mu\_{\rm f}}{\theta\_{\rm f}^2 d\_{\rm s}} + 1.75 \frac{\theta\_{\rm s} \rho\_{\rm f}}{\theta\_{\rm f} d\_{\rm s}} |\mu\_{\rm s} - \mu\_{\rm f}| & \theta\_{\rm f} < 0.8\\ -\frac{3}{4} C\_{\rm D} \frac{\theta\_{\rm s} \rho\_{\rm f}}{d\_{\rm s}} |\mu\_{\rm s} - \mu\_{\rm f}| \theta\_{\rm f}^{-2.65} & \theta\_{\rm f} \ge 0.8 \end{cases} \tag{94}$$
 
$$\left\{ \begin{array}{ll} 24 \left( 1 + 0.15 \operatorname{Re} \frac{0.687}{\mathrm{p}} \right) \\\\ \text{D} \end{array} \right\}\_{\rm D} \tag{1000}$$

$$\mathcal{C}\_{\rm D} = \left\{ \frac{24 \left( 1 + 0.15 \, Re\_{\rm p}^{\rm conv} \right)}{Re\_{\rm p}} \right\} \quad Re\_{\rm p} \le 1000 \tag{95}$$
 
$$\text{Re}\_{\rm p} > 1000$$

*θ*<sup>s</sup> is the particle volume fraction, *θ*<sup>f</sup> is the fluid volume fraction, *μ*<sup>f</sup> is the fluid phase viscosity, *ρ*<sup>f</sup> is the fluid density, *d*<sup>s</sup> is the particle diameter of the solid phase and *C*<sup>D</sup> is the drag coefficient. Particle Reynolds number *Re*<sup>p</sup> is described by

$$Re\_{\rm p} = \theta\_{\rm f} \rho\_{\rm f} d\_{\rm s} |\mathfrak{u}\_{\rm s} - \mathfrak{u}\_{\rm f}|/\mathfrak{u}\_{\rm f} \tag{96}$$

Therefore, Eq. (40), the momentum equation in the x direction, is adjusted by a source term as follows

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial \mathbf{x}} + v \frac{\partial u}{\partial \mathbf{y}} + w \frac{\partial u}{\partial \mathbf{z}} = -\frac{1}{\rho} \frac{\partial p}{\partial \mathbf{x}} + \frac{\mu}{\rho} \left( \frac{\partial^2 u}{\partial \mathbf{x}^2} + \frac{\partial^2 u}{\partial \mathbf{y}^2} + \frac{\partial^2 u}{\partial \mathbf{z}^2} \right) + \mathbf{g}\_{\mathbf{x}} + \mathbf{S}\_{\text{mom},\mathbf{x}} \tag{97}$$

where *S*mom is the source term for momentum which can be expressed as:

$$S\_{\text{mom}, \text{x}} = \sum F\_{\text{D}, i, \text{x}} / (\rho \text{dx} \text{dy} \,\text{dz}) \tag{98}$$

*The Basic Theory of CFD Governing Equations and the Numerical Solution Methods… DOI: http://dx.doi.org/10.5772/intechopen.113253*

Similarly, the momentum equation in the y direction (Eq. 41) and the momentum equation in the z direction (Eq. 42) are adjusted as follows

$$\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} = -\frac{1}{\rho} \frac{\partial p}{\partial y} + \frac{\mu}{\rho} \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right) + \mathbf{g}\_y + \mathbf{S}\_{\text{mom},y} \tag{99}$$

$$\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} = -\frac{1}{\rho} \frac{\partial p}{\partial x} + \frac{\mu}{\rho} \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right) + \mathbf{g}\_x + \mathbf{S}\_{\text{mon},x} \tag{100}$$

#### **3.4 Source term for the energy equation**

As discussed in the energy equation section, the energy equation ensures the thermal energy conservation. In some cases, chemical reactions occur in the fluid domain, and these reactions are either exothermic or endothermic. If the reaction is exothermic, it could release heat to the surrounding fluid and increase the temperature. If the reaction is endothermic, it consumes the heat from the surrounding fluid and decreases the temperature. Even though total energy is conserved, the thermal energy is not. In order to consider the imbalance of thermal energy in fluid, the source term of heat must be included in the energy equation. By considering reaction heat, the energy balance is

$$
\rho \frac{DE}{Dt} \Delta x \Delta y \Delta z = \sum \dot{Q} + r \Delta h \Delta x \Delta y \Delta z \tag{101}
$$

where Δ*h* (J kg�<sup>1</sup> ) is the heat generated when unit mass reactant is converted to products. Bringing Fourier's law of heat conduction and dividing both sides by Δ*x*Δ*y*Δ*z*, the energy equation with source term *S*energ = *r*Δ*h* is

$$
\rho \frac{DE}{Dt} = \lambda \frac{\partial}{\partial \mathbf{x}} \left( \frac{\partial T}{\partial \mathbf{x}} \right) + \lambda \frac{\partial}{\partial \mathbf{y}} \left( \frac{\partial T}{\partial \mathbf{y}} \right) + \lambda \frac{\partial}{\partial \mathbf{z}} \left( \frac{\partial T}{\partial \mathbf{z}} \right) + r \Delta h \tag{102}
$$

With the inclusion of source terms, processes with chemical reactions or multiphase interactions could be accurately simulated in CFD models.
